[note: sorry for the quick and unedited write up on this one. We are a little pressed for time today.]
Got a neat comment from Jim Propp on one of the Aztec diamond posts last night:
“There are other fun problems about counting tilings. The first one I’d give them is counting domino tilings of a 2-by-n rectangle. Then rhombus tilings of an equiangular hexagon with sides of length a,b,1,a,b,1. If the kids like these, let me know; I can suggest some fun follow-ups.”
I thought about the 2 by N rectangle tiling on the way home last night and thought it would, indeed, make a great project to do with the boys today. Despite a little clumsiness keeping the domino tiles on the screen, it was really fun.
I started by introducing the problem and asking the boys to come up with a strategy to solve it. They had some good thoughts:
Following the strategy they came up with in the first part, we started by looking at some simple cases – 2×1, 2×2, 2×3, and 2×4. These cases allowed us to make a few guesses at the pattern:
In the prior video the kids guessed the tiling patter we were seeing 1, 2, 3, 5 was the Fibonacci numbers. Here we looked at the different ways to tile a 2×5 rectangle to see if there were 8.
Sorry we got a little clumsy with the tiles here – we needed a bit more room.
Now that we were feeling a bit more confident that we were seeing the Fibonacci numbers, the question was . . . why? How could we see the Fibonacci pattern in these tilings?
To understand what was going on, we looked to see if we could find a relationship between the combined tilings of the 2×2 and 2×3 recangles and the tilings of 2×4 rectangle. The relationship wasn’t obvious right away, but they did find it!
Sorry again for the clumsiness with the tiles.
Finally, as a last way of double checking the Fibonacci relationship, we looked to see if the tilings of the 2×5 rectangle related to the combined tilings of the 2×3 and 2×4 rectangles.
So, although we didn’t really do a complete formal proof of the relationship between Fibonacci numbers and the tiling patterns, we did hit on the main ideas in that proof. This activity is a great way to introduce some “mathematical thinking” ideas to kids. Thanks to Jim Propp for suggesting it!